Circles - GRE Quantitative Reasoning

Card 0 of 376

Question

What is the area of a circle, one-quarter of the circumference of which is 5.5 inches?

Answer

Here, you need to “solve backward” from the data you have been given. We know that 0.25C = 5.5; therefore, C = 22. In order to solve for the area, we will need the radius of the circle. This can be obtained by recalling that C = 2πr. Replacing 22 for C, we get 22 = 2πr.

Solve for r: r = 22 / 2π = 11 / π.

Now, we solve for the area: A = πr2. Replacing 11 / π for r: A = π (11 / π)2 = (121π) / (π2) = 121 / π.

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Question

Quantitative Comparison

Quantity A: Area of a right triangle with sides 7, 24, 25

Quantity B: Area of a circle with radius 5

Answer

Quantity A: area = base * height / 2 = 7 * 24/2 = 84

Quantity B: area = πr_2 = 25_π

Now we have to remember what π is. Using π = 3, the area is approximately 75. Using π = 3.14, the area increases a little bit, but no matter how exact an approximation for π, this area will never be larger than Quantity A.

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Question

Given circle O with a diameter of 2 and square ABCD inscribed within circle O, what is the area of the shaded region?

Answer

There are two steps to this problem: determining the area of the circle and determining the area of the square. The area of the circle is πr2 which is π(2/1)2 or π. AD is a diameter of circle O and creates two isosceles right triangles with ACD and ABD. The relationship between sides of an isosceles right triangle is 1 : 1 : √2. Thus the sides of square ABCD are √2 and the area is 2. The area of the shaded region is the area of the circle minus the area of the square, or π – 2.

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Question

For $\$15$ , Chelsea can get either a $16\ in$ diameter pizza or two $8\ in$ diameter pizzas. Which is the better deal?

Answer

$A=\pi r^2$

$A_{16\ in}=8^2\pi =64\pi, \ A_{8\ in}=4^2\pi =16\pi$

$2(A_{8\ in})=2(16\pi )=32\pi$

Therefore the 16 inch pizza is the better deal.

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Question

Circle B has a circumference of 36π. What is the area of circle A, which has a radius half the length of the radius of circle B?

Answer

To find the radius of circle B, use the circumference formula (c = πd = 2πr):

2πr = 36π

Divide each side by 2π: r = 18

Now, if circle A has a radius half the length of that of B, A's radius is 18 / 2 = 9.

The area of a circle is πr2. Therefore, for A, it is π*92 = 81π.

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Question

In the diagram above, square ABCD is inscribed in the circle. If the area of the square is 9, what is the area of the circle?

Answer

If the area of the square is 9, then s2 = 9 and s = 3. If the sides thus equal 3, we can calculate the diagonals (either CB or AD) by using the 45-45-90 triangle ratio. For a side of 3, the diagonal will be 3√(2). Note that since the square is inscribed in the circle, this diagonal is also the diameter of the circle. If it is such, the radius is one half of that or 1.5√(2).

Based on that value, we can computer the circle’s area:

A = πr2 = π(1.5√(2))2 = (2.25 * 2)π = 4.5π

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Question

A small circle with radius 5 lies inside a larger circle with radius x. What is the area of the region inside the larger circle, but outside of the smaller circle, in terms of x?

Answer

Since the answers are in terms of pi, simply find the area of each circle in terms of x and ∏:

Smaller: ∏(5)2 = 25∏

Larger: ∏x2

We must subtract the inner circle from the outer circle; this translates to ∏x2-25∏.

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Question

Quantitative Comparison

Quantity A: Area of a circle with radius r

Quantity B: Perimeter of a circle with radius r

Answer

Try different values for the radius to see if a pattern emerges. The formulas needed are Area = π r_2 and Perimeter = 2_πr.

If r = 1, then the Area = π and the Perimeter = 2_π_, so the perimeter is larger.

If r = 4, then the area = 16_π_ and the perimeter = 8_π_, so the area is larger.

Therefore the relationship cannot be determined from the information given.

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Question

Quantitative Comparison

A circle has a radius of 2.

Quantity A: The area of the circle

Quantity B: The circumference of the circle

Answer

This is one of the only special cases where the area equals the circumference of the circle. The Area = πr_2 = 4_π. The circumference = 2_πr_ = 4_π_.

Note: For a quantitative comparison such as this one where the columns have numeric values instead of variables, the answer will rarely be "cannot be determined".

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Question

If a circular garden with a radius of 3 ft. is bordered by a circular sidewalk that is 2 ft. wide, what is the area of the sidewalk?

Answer

To solve this problem, you must find the area of the entire circle (garden and sidewalk) and subtract it by the area of the inner garden. The entire area has a radius of 5 ft. (3 ft. radius of the garden plus the 2 ft. wide sidewalk), giving it an area of $\dpi{100} \small 25\pi$ . The inner garden has a radius of 3 ft. and an area of $\dpi{100} \small 9\pi$ . The difference is $\dpi{100} \small 16\pi$ , which is the area of the sidewalk.

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Question

If a circular monument with a radius of 30 feet is surrounded by a circular garden that is 20 feet wide, what is the area of the garden?

Answer

To find the area of the garden, you need to find the entire area and subtract that by the area of the inner circle, or the monument. The radius of the larger circle is 50, which makes its area $2500\pi$ . The radius of the inner circle is 30, which makes its area $900\pi$ . The difference is $1600\pi$ .

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Question

A given pizza with a -inch diameter has calories. A baker cuts the pizza using a $25^{\circ}$ angle for each piece. If Susan eats five such pieces, how many calories does she consume? Round to the nearest calorie.

Answer

To solve this, notice that one piece of pizza comprises $\frac{25}{360}$ of the total pie or (reducing) $\frac{5}{72}$ of a pie. Now, if Susan buys five slices, she gets:

$\frac{5}{72} \cdot 5 = \frac{25}{72}$ of the pie. If the complete pie contains calories, she will then eat the following amount of calories:

$\frac{25}{72} \cdot 1450 = 503.47$

Rounding, this is calories.

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Question

A circle of radius , for an angle , has an arc length of $3\pi$ . What is the angle ?

Answer

The formula for an arc length, , of a circle of a given radius, , and a given angle, is:

$l=2\pi r(\frac{x}{360})$

Note that $2\pi r$ is the circumference of the circle.

Conversely, for a known arc length and unknown angle, this equation can be rewritten as follows:

$x=360(\frac{l}{2\pi r})$

Plugging in the given values, it is therefore possible to find the missing angle:

$x=360(\frac{3\pi }{2\pi(5 )})=108^{\circ}$

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Question

Find the radius of the given circle:

Answer

To solve this problem, realize that an inscribed angle (an angle formed by two chords) is equal to twice the central angle formed by connecting the origin to the inscribed angle's endpoints:

Now, the formula for an arc length, , of a circle of a given radius, , and a given angle, is:

$l=2\pi r(\frac{x}{360})$

Since the radius is the unknown, this equation can be rewritten as:

$r=(\frac{360}{x})(\frac{l}{2\pi })$

Plugging in our values, we find:

$r=\frac{360}{60}\frac{6\pi}{2\pi}=18$

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Question

What is the circumference of a circle with an area of 36π?

Answer

We know that the area of a circle can be expressed: a = πr2

If we know that the area is 36π, we can substitute this into said equation and get: 36π = πr2

Solving for r, we get: 36 = r2; (after taking the square root of both sides:) 6 = r

Now, we know that the circuference of a circle is expressed: c = πd. Since we know that d = 2r (two radii, placed one after the other, make a diameter), we can rewrite the circumference equation to be: c = 2πr

Since we have r, we can rewrite this to be: c = 2π*6 = 12π

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Question

Circle A has an area of $121\pi$ . What is the perimeter of an enclosed semi-circle with half the radius of circle A?

Answer

Based on our information, we know that the 121π = πr2; 121 = r2; r = 11.

Our other circle with half the radius of A has a diameter equal to the radius of A. Therefore, the circumference of this circle is 11π. Half of this is 5.5π. However, since this is a semi circle, it is enclosed and looks like this:

Therefore, we have to include the diameter in the perimeter. Therefore, the total perimeter of the semi-circle is 5.5π + 11.

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Question

Which is greater: the circumference of a circle with an area of $25\pi \ in^2$ , or the perimeter of a square with side length inches?

Answer

Starting with the circle, we need to find the radius in order to get the circumference. Find $\dpi{100} \small r$ by plugging our given area into the equation for the area of a circle:

$A = \pi r^2$

$25\pi = \pi r^2$

$25 = r^2$

$r = 5\ in$

Then calculate circumference:

$C = 2\pi r$

$\dpi{100} \small C = 2\pi \times 5 = 10\pi \approx 31.4 inches$ (approximating $\dpi{100} \small \pi$ as 3.14)

To find the perimeter of the square, we can use $P = 4s$ , where $\dpi{100} \small P$ is the perimeter and $\dpi{100} \small s$ is the side length:

$P = 4 \times 7\ in = 28\ in$

$\dpi{100} \small 31.4>28$ , so the circle's circumference is greater.

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Question

$x\ge 1$

Quantity A: The circumference of a circle with radius

Quantity B: The area of a circle with a diameter one fourth the radius of the circle in Quantity A

Which of the following is true?

Answer

Let's compute each value separately. We know that the radii are positive numbers that are greater than or equal to . This means that we do not need to worry about the fact that the area could represent a square of a decimal value like .

Quantity A

Since $c=2\pi r$ , we know:

$a=2 \cdot \pi \cdot 24x = 48x\pi$

Quantity B

If the diameter is one-fourth the radius of A, we know:
$d = \frac{1}{4}\cdot 24x = 6x$

Thus, the radius must be half of that, or .

Now, we need to compute the area of this circle. We know:

$A = \pi r^2$

Therefore, $A = \pi \cdot (3x)^2 = 9x^2$

Now, notice that if , Quantity A is larger.

However, if we choose a value like , we have:

Quantity A: $48 * 1000 \pi = 48000\pi$

Quantity B: $9 * 1000^2\pi= 9000000\pi$

Therefore, the relation cannot be determined!

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Question

Circle has a center in the center of Square .

The area of Square is .

What is the circumference of Circle ?

Answer

Since we know that the area of Square is , we know , where is the length of one of its sides. From this, we can solve for by taking the square root of both sides. You will have to do this by estimating upward. Therefore, you know that is . By careful guessing, you can quickly see that is . From this, you know that the diameter of your circle must be half of , or (because it is circumscribed). Therefore, you can draw:

The circumference of this circle is defined as:

$C=2\pi r$ or, for your values:

$c=2\pi * 12 = 24\pi$

(You could also compute this from the diameter, but many students just memorize the formula above.)

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Question

John owns 8 black shirts, 7 red shirts, 6 blue shirts and 4 white shirts. If he wants to make a circle chart of his shirts, what is the degree angle corresponding to the "blue shirt section?"

Answer

We can set up a ratio to calculate the angle measure as such: 6/25 = x/360, since there are 360 degrees in a circle. Solving, we obtain x = 86.4 degrees.

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